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Sinc-type functions on a class of nilpotent Lie groups

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Veröffentlicht/Copyright: 14. Januar 2014
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Advances in Pure and Applied Mathematics
Aus der Zeitschrift Advances in Pure and Applied Mathematics Band 5 Heft 1

Abstract.

Let N be a simply connected, connected nilpotent Lie group with the following assumptions. Its Lie algebra 𝔫 is an n-dimensional vector space over the reals. Moreover, 𝔫=𝔷⊕𝔟⊕𝔞, 𝔷 is the center of 𝔫, 𝔷=ℝZn-2d⊕ℝZn-2d-1⊕⋯⊕ℝZ1, 𝔟=ℝYd⊕ℝYd-1⊕⋯⊕ℝY1, 𝔞=ℝXd⊕ℝXd-1⊕⋯⊕ℝX1. Next, assume 𝔷⊕𝔟 is a maximal commutative ideal of 𝔫, [𝔞,𝔟]⊆𝔷, and det ([Xi,Yj])1≤i,j≤d is a non-trivial homogeneous polynomial defined over the ideal [𝔫,𝔫]⊆𝔷. We do not assume that [𝔞,𝔞] is generally trivial. We obtain some precise description of band-limited spaces which are sampling subspaces of L2(N) with respect to some discrete set Γ. The set Γ is explicitly constructed by fixing a strong Malcev basis for 𝔫. We provide sufficient conditions for which a function f is determined from its sampled values on (f(γ))γ∈Γ. We also provide an explicit formula for the corresponding sinc-type functions. Several examples are also computed in the paper.

Keywords: Sampling; interpolation; nilpotent Lie groups; representations
MSC: 22E25; 22E27

Funding source: Czech Ministry of Education

Award Identifier / Grant number: ERC CZ LL1203

The author thanks the anonymous reviewer for a careful and thorough reading. His suggestions and corrections greatly improved the quality of the paper.

Received: 2013-08-19
Revised: 2013-12-20
Accepted: 2013-12-20
Published Online: 2014-01-14
Published in Print: 2014-03-01

© 2014 by Walter de Gruyter Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. A note on generalized absolute Cesàro summability
  3. Sinc-type functions on a class of nilpotent Lie groups
  4. Some new general integral inequalities for h-convex and h-concave functions
  5. The construction of periodic unfolding operators on some compact Riemannian manifolds
  6. Integration operators from the space of Cauchy integral transforms to the Dirichlet space
https://doi.org/10.1515/apam-2013-0028
Schlagwörter für diesen Artikel
Sampling; interpolation; nilpotent Lie groups; representations

Artikel in diesem Heft

  1. Frontmatter
  2. A note on generalized absolute Cesàro summability
  3. Sinc-type functions on a class of nilpotent Lie groups
  4. Some new general integral inequalities for h-convex and h-concave functions
  5. The construction of periodic unfolding operators on some compact Riemannian manifolds
  6. Integration operators from the space of Cauchy integral transforms to the Dirichlet space
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